where \(\mu _i\)  is the weight associated with
\(\mathcal {R}_i\) . In the case where \(\mathcal {R}_1,..., \mathcal {R}_l\)  are discrete measures with finite number of elements and the weights in \(\mu \)  are uniform, it
is shown by [1]} that the problem of exploring Wasserstein barycenter \(\tilde{\mathcal {R}}_{l,\mu }\)  on the space of \(S_2(\theta )\) 
in (4) is recast to search only on \(\mathcal {O}_r(\theta )\)  denoting as a set of probability measures
with at most \(r\)  support points in \(\theta \) , where \(r= \sum _{i=1}^l e_i-l+1\)  and \(e_i\)  is the number of
elements in \(\mathcal {R}_i\)  for all \(1 \le i \le l\) . Moreover, an efficient algorithm for exploring local solutions of the Wasserstein barycenter problem over \(\mathcal {O}_r(\theta )\)  for some \(r \ge 1\)  has been studied by [2]}.
